| L(s) = 1 | − 6.73e8·2-s + 5.51e13·3-s + 3.09e17·4-s − 1.13e20·5-s − 3.71e22·6-s + 6.16e23·7-s − 1.11e26·8-s + 1.46e27·9-s + 7.63e28·10-s − 4.46e28·11-s + 1.70e31·12-s − 1.50e31·13-s − 4.15e32·14-s − 6.25e33·15-s + 3.02e34·16-s + 1.30e35·17-s − 9.88e35·18-s + 2.11e36·19-s − 3.50e37·20-s + 3.39e37·21-s + 3.00e37·22-s − 1.21e39·23-s − 6.12e39·24-s + 5.92e39·25-s + 1.01e40·26-s − 5.61e39·27-s + 1.90e41·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 1.39·3-s + 2.14·4-s − 1.36·5-s − 2.46·6-s + 0.506·7-s − 2.03·8-s + 0.935·9-s + 2.41·10-s − 0.0933·11-s + 2.98·12-s − 0.269·13-s − 0.898·14-s − 1.89·15-s + 1.45·16-s + 1.11·17-s − 1.65·18-s + 0.761·19-s − 2.92·20-s + 0.705·21-s + 0.165·22-s − 1.88·23-s − 2.82·24-s + 0.854·25-s + 0.477·26-s − 0.0903·27-s + 1.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(58-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+57/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(29)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{59}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 6.73e8T + 1.44e17T^{2} \) |
| 3 | \( 1 - 5.51e13T + 1.57e27T^{2} \) |
| 5 | \( 1 + 1.13e20T + 6.93e39T^{2} \) |
| 7 | \( 1 - 6.16e23T + 1.48e48T^{2} \) |
| 11 | \( 1 + 4.46e28T + 2.28e59T^{2} \) |
| 13 | \( 1 + 1.50e31T + 3.12e63T^{2} \) |
| 17 | \( 1 - 1.30e35T + 1.36e70T^{2} \) |
| 19 | \( 1 - 2.11e36T + 7.74e72T^{2} \) |
| 23 | \( 1 + 1.21e39T + 4.15e77T^{2} \) |
| 29 | \( 1 + 7.31e41T + 2.27e83T^{2} \) |
| 31 | \( 1 + 1.88e42T + 1.01e85T^{2} \) |
| 37 | \( 1 - 3.72e44T + 2.44e89T^{2} \) |
| 41 | \( 1 + 2.74e45T + 8.48e91T^{2} \) |
| 43 | \( 1 + 3.58e46T + 1.28e93T^{2} \) |
| 47 | \( 1 + 7.49e47T + 2.03e95T^{2} \) |
| 53 | \( 1 + 1.28e49T + 1.92e98T^{2} \) |
| 59 | \( 1 - 1.82e50T + 8.68e100T^{2} \) |
| 61 | \( 1 + 3.13e50T + 5.80e101T^{2} \) |
| 67 | \( 1 + 6.22e51T + 1.21e104T^{2} \) |
| 71 | \( 1 - 3.91e52T + 3.32e105T^{2} \) |
| 73 | \( 1 + 3.35e51T + 1.61e106T^{2} \) |
| 79 | \( 1 - 3.32e53T + 1.46e108T^{2} \) |
| 83 | \( 1 - 7.49e53T + 2.44e109T^{2} \) |
| 89 | \( 1 - 4.16e55T + 1.30e111T^{2} \) |
| 97 | \( 1 - 3.60e56T + 1.76e113T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.45450330181263702795377678909, −16.21614508102757385593189497225, −14.78734617927403683350184652033, −11.60913412900916759381674449433, −9.655328233176605859002595396270, −8.084656797984681345198693411590, −7.67748387239562890329726998409, −3.43596312250581772988677705601, −1.76084178090973606619894531008, 0,
1.76084178090973606619894531008, 3.43596312250581772988677705601, 7.67748387239562890329726998409, 8.084656797984681345198693411590, 9.655328233176605859002595396270, 11.60913412900916759381674449433, 14.78734617927403683350184652033, 16.21614508102757385593189497225, 18.45450330181263702795377678909
